APPENDIX III. LEAST SQUARES WITH ERRORS IN BOTH VARIABLES

Experiments in physics designed to determine parameters
in the functional relationship between quantities x and y
involve a series of measurements of x and the corresponding y.
In many cases not only are there measurement errors
yi for
each yj, but also measurement errors
xj for
each xj. Most
physicists treat the problem as if all the
xj = 0
using the
standard least squares method. Such a procedure loses accuracy
in the determination of the unknown parameters contained in
the function y = f (x) and it gives estimates of
errors which are smaller than the true errors.

The standard least squares method of Section 15 should be
used only when all the
xj <<
yi.
Otherwise one must replace the weighting factors
1 / i2
in Eq. (24) with
(j)-2
where

We see that the standard least squares computer programs may
still be used. In the case where
y = 1 +
2x one may use
what
are called linear regression programs, and where y is a polynomial
in x one may use multiple polynomial regression programs.
The usual procedure is to guess starting values for
ðf / ð x and then solve for the parameters
j* using Eq. (30)
with
j replaced by
j. Then new
[ðf / ð x]j can
be evaluated and the procedure repeated.
Usually only two iterations are necessary. The effective
variance method is exact in the limit that
ðf / ð x is constant over the region
xj. This
means it is always exact for linear regressions.